205553: Fourier Analysis

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20-5553 Fourier Analysis

• Analogue Filters

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An example of a low pass filter One simple low
pass filter may be described by the differential
equation
.
is the input,
is the output,
where
and a is a constant.
To see the effect this has on Fourier components
of different frequencies, consider the effect on
an input
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Use Laplace Transform method of solution
Re-arrange to get
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Get the solution from the inverse transform
Now actually we are only interested in the steady
state response, so we can disregard the
exponential terms, since they approach zero as t
approaches infinity
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To investigate this function , we need to obtain
the amplitude and phase - write it in the form
Expand the sin term
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Comparing coefficients
so
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Substituting for R, we therefore find that the
steady state solution to an input of
is
The effect is that the amplitude is reduced, by a
factor proportional to
There is also a phase shift of
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Plot the amplitude ratio and phase against
frequency (take a as 1)
Waves of low frequency pass through relatively
unchanged, but high frequency terms are
successively more attenuated as the frequency
increases.
Lowpass filter